Or can I only use it from my phone? I downloaded and installed bitcoin core. took a day to get up to date. But wallet? Balance is zero. How do or must I log in? How does it work anyway?
In general, each wallet has its own set of addresses and private key chain. There is no way to log into your Bitcoin wallet, as there is no central service that is providing it to you, rather it is just locally stored on your device.
I would not recommend trying to share the same set of keys between different wallets, as it would increase the attack vectors on your money.
Rather, you might want to just send most of your bitcoins to whichever storage you expect to be most secure, and hold a spending amount on the other devices.
If you just want to see the coins on other devices, you may look up the specific address on a blockchain explorer such as blockchain.info, or read up on "watch only wallet".